Most problems encountered in engineering design are nonlinear by nature and involve the determination of system parameters that satisfy certain goals for the problem being solved. Such problems can be cast in the form of a mathematical optimization problem where a solution is desired that minimizes a system function or parameter subject to limitations or constraints on the system. Both the system function and constraints are comprised of system inputs (control variables) and system outputs, which may be either discrete or continuous. Furthermore, constraints may be equalities or inequalities. The solution to a given optimization problem has either or both of the following characteristics: 1) minimizes or maximizes a desired condition or conditions, thus satisfying the optimality condition and 2) satisfies the set of constraint equations imposed on the system.
With the above definitions, several categories of optimization problems may be defmed. A Free Optimization Problem (FOP) is one for which no constraints exist. A Constraint Optimization Problem (COP) includes both constraints and a minimize (or maximize) condition(s) requirement. In contrast, a Constraint Satisfaction Problem (CSP) contains only constraints. Solving a CSP means finding one feasible solution within the search space that satisfies the constraint conditions. Solving a COP means finding a solution that is both feasible and optimal in the sense that a minimum (or maximum) value for the desired condition(s) is realized.
The solution to such a problem typically involves a mathematical search algorithm, whereby successively improved solutions are obtained over the course of a number of algorithm iterations. Each iteration, which can be thought of as a proposed solution, hopefully results in improvement of an objective function. An objective function is a mathematical expression having parameter values of a proposed solution as inputs. The objective function produces a figure of merit for the proposed solution. Comparison of objective function values provides a measure as to the relative strength of one solution versus another. Numerous search algorithms exist and differ in the manner by which the control variables for a particular problem are modified, whether a population of solutions or a single solution is tracked during the improvement process, and the assessment of convergence. However, these search algorithms rely on the results of an objective function in deciding a path of convergence. Examples of optimization algorithms include Genetic Algorithms, Simulated Annealing, and Tabu Search.
Within optimization algorithms, the issue of handling constraints for COPs and CSPs must be addressed. Several classes of methods exist for dealing with constraints. The most widespread method is the use of the penalty approach for modifying the objective function, which has the effect of converting a COP or CSP into a FOP. In this method, a penalty function, representing violations in the set of constraint equations, is added to an objective function characterizing the desired optimal condition. When the penalty function is positive, the solution is infeasible. When the penalty function is zero, all constraints are satisfied. Minimizing the modified objective function thus seeks not only optimality but also satisfaction of the constraints.
For a given optimization search, the penalty approach broadens the search space by allowing examination of both feasible and infeasible solutions in an unbiased manner. Broadening the search space during an optimization search often allows local minima to be circumnavigated more readily, thus making for a more effective optimization algorithm. In contrast, alternate methods for handling constraints, such as infeasible solution ‘repair’ and ‘behavioral memory’, are based on maintaining or forcing feasibility among solutions that are examined during the optimization search.
To implement the penalty approach, a mathematical expression is defined for each constraint that quantifies the magnitude of the constraint violation. For the given constraint, a weighting factor then multiplies the result to create an objective function penalty component. Summing all penalty components yields the total penalty. The larger the weighting factor for a given constraint, the greater the emphasis the optimization search will place on resolving violations in the constraint during the optimization search. Many approaches exist for defining the form of the penalty function and the weighting factors. As defined by the resultant modified objective function, weighting factors are problem specific and are bounded by zero (the constraint is not active) and infinity (the search space omits all violations of the constraint).
The simplest penalty function form is the ‘death penalty’, which sets the value of the weighting factor for each constraint to infinity. With a death penalty the search algorithm will immediately reject any violation of a constraint, which is equivalent to rejecting all infeasible solutions. Static penalties apply a finite penalty value to each constraint defined. A static weighting factor maintains its initial input value throughout the optimization search. Dynamic penalties adjust the initial input value during the course of the optimization search according to a mathematical expression that determines the amount and frequency of the weight change. The form of the penalty functions in a dynamic penalty scheme contains, in addition to the initial static penalty weighting factors (required to start the search), additional parameters that must be input as part of the optimization algorithm.
Similar to dynamic penalties, adaptive penalties adjust weight values over the course of an optimization search. In contrast, the amount and frequency of the weight change is determined by the progress of the optimization search in finding improved solutions. Several approaches for implementing adaptive penalty functions have been proposed. Bean and Hadj-Alouane created the method of Adaptive Penalties (AP), which was implemented in the context of a Genetic Algorithm. In the AP method, the population of solutions obtained over a preset number of iterations of the optimization search is examined and the weights adjusted depending on whether the population contains only feasible, infeasible, or a mixture of feasible and infeasible solutions. Coit, Smith, and Tate proposed an adaptive penalty method based on estimating a ‘Near Feasibility Threshold’ (NFT) for each given constraint. Conceptually, the NFT defines a region of infeasible search space just outside of feasibility that the optimization search would then be permitted to explore. Eiben and Hemert developed the Stepwise Adaption of Weights (SAW) method for adapting penalties. In their method, a weighting factor adjustment is made periodically to each constraint that violates in the best solution, thus potentially biasing future solutions away from constraint violations.
Several deficiencies exist in the penalty methods proposed. Death penalties restrict the search space by forcing all candidate solutions generated during the search to satisfy feasibility. In the static weighting factor approach, one must perform parametric studies on a set of test problems that are reflective of the types of optimization applications one would expect to encounter, with the result being a range of acceptable weight values established for each constraint of interest. The user would then select the weight values for a specific set of constraints based on a pre-established range of acceptable values. Particularly for COPs, varying the static weight values for a given problem can often result in a more or less optimal result. Similarly, dynamic penalties require the specification of parameters that must be determined based on empirical data. Fine-tuning of such parameters will often result in a different optimal result.
Penalty adaptation improves over the static and dynamic penalty approaches by attempting to utilize information about the specific problem being solved as the optimization search progresses. In effect, the problem is periodically redefined. A deficiency with the adaptive penalty approach is that the objective function loses all meaning in an absolute sense during the course of an optimization search. In other words, there is no ‘memory’ that ties the objective function back to the original starting point of the optimization search as exists in a static penalty or dynamic penalty approach.
One known optimization problem involves design of an operation strategy for a nuclear reactor such as a boiling water nuclear reactor. FIG. 15 illustrates a conventional boiler water reactor. As shown, a jet pump 110 supplies water to a reactor vessel 112 housed within a containment vessel 114. The core 116 of the reactor vessel 112 includes a number of fuel bundles such as described in detail below with respect to FIG. 16. The controlled nuclear fission taking place at the fuel bundles in the core 116 generates heat which turns the supplied water to steam. This steam is supplied from the reactor vessel to turbines 118, which power a generator 120. The generator 120 then outputs electrical energy. The steam supplied to the turbines 118 is recycled by condensing the steam back into water at a condenser 122, and supplying the condensed steam back to the jet pump 110.
FIG. 16 illustrates a fuel bundle in the core 116 of the reactor vessel 112. A typical core will contain anywhere from 200 to 900 of these bundles B. As shown in FIG. 16 the bundle B includes an outer channel C surrounding a plurality of fuel rods 100 extending generally parallel to one another between upper and lower tie plates U and L, respectively, and in a generally rectilinear matrix of fuel rods as illustrated in FIG. 17. The rods 100 are maintained laterally spaced from one another by a plurality of spacers S vertically spaced from the other along the length of the fuel rods within the channel C. Referring to FIG. 17, there is illustrated in an array of fuel rods 100, i.e., in this instance, a 10×10 array, surrounded by the fuel channel C. The fuel rods 100 are arranged in orthogonally related rows and also surround one or more water rods, two water rods 130 being illustrated. The fuel bundle B is arranged in one quadrant of a control rod or blade 132 as is conventional. It will be appreciated that a fuel bundle is typically arranged in each of the other quadrants of the control blade 132. Movement of the control blade 132 up between the bundles B controls the amount of reactivity occurring in the bundles B in association with that control blade 132.
A nuclear reactor core includes many individual components that have different characteristics that may affect a strategy for efficient operation of the core. For example, a nuclear reactor core has many, e.g., several hundred, individual fuel assemblies (bundles) that have different characteristics and which must be arranged within the reactor core or “loaded” so that the interaction between fuel bundles satisfies all regulatory and reactor design constraints, including governmental and customer specified constraints. Similarly, other controllable elements and factors that affect the reactivity and overall efficiency of a reactor core must also be taken into consideration if one is to design or develop an effective control strategy for optimizing the performance of a reactor core at a particular reactor plant. Such “operational controls” (also referred to interchangeably herein as “independent control-variables” and “design inputs”) include, for example, various physical component configurations and controllable operating conditions that can be individually adjusted or set.
Besides fuel bundle “loading”, other sources of control variables include “core flow” or rate of water flow through the core, the “exposure” and the “reactivity” or interaction between fuel bundles within the core due to differences in bundle enrichment, and the “rod pattern” or distribution and axial position of control blades within the core. As such, each of these operational controls constitutes an independent control-variable or design input that has a measurable effect on the overall performance of the reactor core. Due to the vast number of possible different operational values and combinations of values that these independent control-variables can assume, it is a formidable challenge and a very time consuming task, even using known computer-aided methodologies, to attempt to analyze and optimize all the individual influences on core reactivity and performance.
For example, the number of different fuel bundle configurations possible in the reactor core can be in excess of one hundred factorial. Of the many different loading pattern possibilities, only a small percentage of these configurations will satisfy all of the requisite design constraints for a particular reactor plant. In addition, only a small percentage of the configurations that satisfy all the applicable design constraints are economically feasible.
Moreover, in addition to satisfying various design constraints, since a fuel bundle loading arrangement ultimately affects the core cycle energy (i.e., the amount of energy that the reactor core generates before the core needs to be refueled with new fuel elements), a particular loading arrangement needs to be selected that optimizes the core cycle energy.
In order to furnish and maintain the required energy output, the reactor core is periodically refueled with fresh fuel bundles. The duration between one refueling and the next is commonly referred to as a “fuel-cycle” or “core-cycle” of operation and, depending on the particular reactor plant, is on the order of twelve to twenty-four (typically eighteen) months. At the time of refueling, typically one third of the least reactive fuel are removed from the reactor and the remaining fuel bundles are repositioned before fresh fuel bundles are added. Generally, to improve core cycle energy higher reactivity bundles should be positioned at interior core locations. However, such arrangements are not always possible to achieve while still satisfying plant specific design constraints. Since each fuel bundle can be loaded at a variety of different locations relative to other bundles, identifying a core loading arrangement that produces optimum performance of the core for each fuel-cycle presents a complex and computation-intensive optimization problem that can be very time consuming to solve.
During the course of a core-cycle, the excess energy capability of the core, defined as the excess reactivity or “hot excess”, is controlled in several ways. One technique employs a burnable reactivity inhibitor, e.g., Gadolinia, incorporated into the fresh fuel. The quantity of initial burnable inhibitor is determined by design constraints and performance characteristics typically set by the utility and by the Nuclear Regulatory Commission (NRC). The burnable inhibitor controls most, but not all, of the excess reactivity. Consequently, “control blades” (also referred to herein as “control rods”)—which inhibit reactivity by absorbing nuclear emissions—are also used to control excess reactivity. Typically, a reactor core contains many such control blades which are fit between selected fuel bundles and are axially positionable within the core. These control blades assure safe shut down and provide the primary mechanism for controlling the maximum power peaking factor.
The total number of control blades utilized varies with core size and geometry, and is typically between 50 and 150. The axial position of the control blades (e.g., fully inserted, fully withdrawn, or somewhere in between) is based on the need to control the excess reactivity and to meet other operational constraints, such as the maximum core power peaking factor. For each control blade, there may be, for example, 24, 48 or more possible axial positions or “notches” and 40 “exposure” (i.e., duration of use) steps. Considering symmetry and other requirements that reduce the number of control blades that are available for application at any given time, there are many millions of possible combinations of control blade positions for even the simplest case. Of these possible configurations, only a small fraction satisfies all applicable design and safety constraints, and of these, only a small fraction is economical. Moreover, the axial positioning of control blades also influences the core cycle energy that any given fuel loading pattern can achieve. Since it is desirable to maximize the core-cycle energy in order to minimize nuclear fuel cycle costs, developing an optimum control blade positioning strategy presents another formidable independent control-variable optimization problem that must also be taken into consideration when attempting to optimize fuel-cycle design and management strategies.
Core design and the development of an operation strategy typically involves a constraint optimization problem wherein a best possible solution that maximizes energy output is developed according to various well-known algorithms. For example, a reactor core and operating strategy may be designed to generate a certain amount of energy measured in gigawatt days per metric ton or uranium (GWD/MTU) over a cycle before being replaced with a new core.
As discussed above, developing a solution to such a constraint problem typically involves a mathematical search algorithm, whereby successively improved solutions are obtained over the course of a number of algorithm iterations. Each iteration, which can be thought of as a proposed solution, hopefully results in improvement of an objective function, producing a figure of merit for the proposed solution. Comparison of objective function values provides a measure as to the relative strength of one solution versus another. Numerous search algorithms for core and operational strategy design exist that rely on the results of an objective function in deciding a path of convergence.
At the beginning of cycle (BOC), the core design is put into operation. As is also typical, actual reactor performance often deviates from the performance modeled in generating the core design. Adjustments from the operational model are quite often made in order to maintain performance of the reactor before the end of cycle (EOC). Accordingly, the desire for robustness in a design solution arises from the fact that the assumptions that form the basis of a given design may change once the plant starts operating. Assumptions fall into several categories. First, there are the assumed operational conditions of the plant, which include for example, the power level, flow, and inlet temperature. Second, there are the assumed biases in the simulation model that are based on historical data. As is known, developing a core and/or operational strategy design solution involves running simulations of the reactor using a proposed solution and using outputs from the simulation as inputs to an objective function, which provides a figure of merit for the propose solution. Numerous simulation programs for simulating reactor performance are known in the art. An example of a simulation model bias is the core eigenvalue, which is a measure of core reactivity or neutron balance, at hot and cold conditions as function of cycle exposure (for a critical core the eigenvalue should be 1.00 but typically ranges between 0.99 and 1.01).
Another category of assumption is assumed margins in the simulation model for each of the thermal and reactivity parameters. Design margins are introduced to account for uncertainties in the simulation model and to assure that once the plant starts operating, thermal and reactivity limits are not violated (the so-called operating margin). Examples of thermal parameters are MFLPD, MFLPCR, and MAPRAT. Examples of reactivity parameters are cold shutdown margin and hot excess reactivity. Reactivity limits include cold shutdown margin (CSDM) and hot excess reactivity (HOTX). CSDM is defined as the reactivity margin to the limit for the reactor in a cold state, with all control blades inserted with the exception of the most reactive control blade. CSDM is determined for each time (exposure) state-point during the cycle. HOTX is defined as the core reactivity for the reactor in a hot state, with all control blades removed, for each exposure state-point during the cycle. Thermal limits include MFLPD (Maximum Fraction of Limiting Power Density), MAPRAT (the ratio of MAPLHGR or Maximum Average Planar Linear Heat Generation compared to its limit), and MFLCPR (Maximum Fraction of Limiting Critical Power Ratio). MFLPD is defined as the maximum of the ratio of local rod power or linear heat generation rate (i.e. kiloWatts per unit length) in a given bundle at a given elevation, as compared to the limiting value. MAPLHGR is the maximum average linear heat generation rate (LHGR) over the plane in a given bundle at a given elevation. MAPRAT is simply the ratio of MAPLHGR to the limiting value. LHGR limits protect the fuel against the phenomena of fuel cladding plastic strain, fuel pellet centerline melting, and lift-off, which is bulging of the clad that exceeds the expansion of the pellet due primarily to fission gas build-up. Lift-off degrades the heat transfer from the pellet across the clad to the coolant. MAPRAT limits protect the fuel during the postulated loss of coolant accident while MFLPD limits protect the fuel during normal operation. MFLCPR protects the fuel against the phenomena of ‘film dryout’. In BWR (boiling water reactor) heat transfer, a thin film of water on the surface of the fuel rod assures adequate removal of the heat generated in the fuel rod as water is converted into steam. This mechanism, also known as nucleate boiling, will continue as the power in the fuel rod is increased up until a point known as transition boiling. During transition boiling, heat transfer degrades rapidly leading to the elimination of the thin film and ultimately film dryout, at which time the cladding surface temperature increases rapidly leading to cladding failure. The Critical Power of the bundle is the power at which a given fuel bundle achieves film dryout, and is determined from experimental tests. The Critical Power Ratio (CPR) is the ratio of the critical power to the actual bundle power. MFLCPR is simply the maximum over all bundles of the fraction of each bundles CPR to the limiting value.
Operating margins may be communicated to a core monitoring system, and are derived from plant measurement or instrumentation system. In a BWR, the instrumentation system is comprised of fixed detectors and removable detectors. The removable detectors, or TIPS (traversing in-core probes), are inserted each month to calibrate the fixed detectors. This is due to the fact that the fixed detectors will ‘burn out’ due to the neutron environment and so must have their signals adjusted. As will be appreciated, however, in a simulator the measurements are simulated. A loss of operating margin may require adjustment of the control blade pattern and/or core flow in order to redistribute the power. The control blade pattern is the amount by which each of the control blades is inserted into the core and how these positions are planned to change over time. Core flow is the flow of water through the core.
Changes in any of the design assumptions—operational conditions, model biases, or margins—may require changes in the reactor control parameters, once the plant begins operation. Avoidance of abrupt changes in core output response (e.g. local power) due to a required change in one of the control variables (e.g. control blade notch) is important from the perspective of plant safety as well as ease of operation.
Core design is currently performed using a fixed set of assumptions. This method of design does not provide information as to the robustness of a given solution. A design may satisfy all design margins for the input set of assumption but may prove to have reduced margins (or worse, approach violations in thermal or reactivity limits) during plant operation. In such instances, the reactor operators would modify the operational strategy (control blade placements and core flow) to recover the lost margin. Typically, such modifications to the operational strategy would be first simulated using the on-line predictive capabilities of the core monitoring system, beginning with a ‘snapshot’ of the plant state based on the plant measurement and operating conditions. During the simulation of these various scenarios, the degree of robustness of the current solution will become evident. A solution that is brittle may require additional operational maneuvering (such as use of an alternate set of control blades) in order to achieve a robust solution. This maneuvering may require a reduction in core power (and lost electrical generation) during the ‘transition’ maneuver to the new core state.
An alternative method is to perform a simulation of the base design with a single change in one of the design parameters and validate that a success path, involving a change in operational strategy, exists for satisfying the thermal and reactivity limits. For example, one could change the target hot eigenvalue from 1.0 to 1.003 over the cycle and manually perturb control blades and core flow within the simulation to satisfy thermal and reactivity limits. If no such success path existed, it would be necessary to change the overall design. Examples of such changes would be to perform fuel shuffles, utilize a different set of control blades (e.g. an A1 sequence vs. an A2 sequence), or modify the fresh bundle design. This process is extremely time consuming and can only examine singular changes in the design parameters.